Skip to main content

Full text of "Analytical Mechanics"

See other formats


EQUILIBRIUM OF FLEXIBLE CORDS
63
PROBLEM. A bridge is supported by two suspension cables. The bridge has a weight of 1.5 tons per horizontal foot and has a span of 400 feet. Supposing the dip of the bridge to be 50 feet find the values of the tensile force at the lowest and highest points of the cable.
64. Equilibrium of a Uniform Flexible Cord which is Suspended from Its Ends.  The problem is to determine the nature of the curve which a perfectly uniform and flexible cable will assume when suspended from two points. Let AOB, Fig. 42, be the curve. Consider the equilibrium of that part of the cable which is between the lowest point 0 and any other point P. The part of the cable which is under consideration is acted upon by the following three forces:
The tensile force at the point 0, T0.
The tensile force at the point P, T.
The weight of the cable between the points 0 and P. Since the cable is perfectly flexible T0 and T are tangent to the curve.   Therefore we have
- To + T cos 6 = 0,    or  T cos 0 = TO,            (1)
0=0,    or  rsin0 = ws,           (2)
x
where w is the weight per unit length of the cable and s is the length of OP.
Squaring equations (1) and (2) and adding we obtain
r2=2V + wV.                            (3)
Eliminating T between equations (1) and (2) we get
(4)
m
=  tan 0,
w         '
which is the intrinsic equation of the curve.